Method of estimating transmission channel response and difference of synchronization offsets introduced in a received stream of packets of OFDM data and relative receiver

ABSTRACT

An OFDM receiver includes a sampling circuit configured to sample an incoming signal received through a transmission channel and an estimation circuit configured to receive samples of the incoming signal and to estimate transmission channel response and eventual differences of synchronization offsets introduced at a receiver side. An equalizer may be coupled to the estimation circuit and configured to compensate an effect of the transmission channel response and of the differences of synchronization offsets on the received samples and to generate equalized samples. An OFDM detector may be configured to generate a stream of demodulated digital symbols based upon the equalized samples.

FIELD OF THE INVENTION

This invention relates to digital communications and, more particularly,to a method of estimating transmission channel response and differenceof synchronization offsets introduced in a received stream of packets ofOFDM data and a relative receiver of OFDM digital symbols.

BACKGROUND OF THE INVENTION

High-speed communication systems use a relatively large bandwidth toobtain data rates up to hundreds of mega bits per second. With referenceto wireless and power-line systems, in such bandwidths the frequencyselective nature of the channel may limit the overall systemperformance.

A robust modulation against the frequency-selectivity of the channel isorthogonal frequency division multiplexing (OFDM), which transforms afrequency selective channel into a set of parallel flat sub-channels. Inthis context, the so-called channel estimation, i.e. the estimation offunctioning parameters of the transmission channel, is a crucial elementto demodulate the received data. Considering packet communications, aknown header is transmitted at the beginning of each packet and may beused to carry out data-aided channel estimations. The accuracy of thisestimation depends on the number of symbols of the header, which isgenerally small to reduce the overhead of the packet.

Assuming the channel time-invariant during the transmission of severalpackets, a method to improve channel estimation may be to average outthe estimates performed during previous packets. The phase of theestimated channel linearly depends on both the frame synchronizationpoint and the phase of the sampling clock, hence it may be differentfrom packet to packet. This implies that the channel estimations shouldnot be simply averaged out, but, in order to have a reliable estimation,linear phase differences may be estimated and compensated.

This issue has been previously tackled in “Improved HomePlug AV channelestimation exploiting sounding procedure,” Riva, M. Odoni, E. Guerriniand P. Bisaglia, IEEE ISPLC 2009, pp. 296-300, 2009 and in “ImprovedOFDM channel estimation using inter-packet information,” D. Fu, IEEEACSSC 2005, pp. 514-518, October 2005. Unfortunately, the techniquedisclosed in Riva at al. performs well when the phase of the samplingclock does not change from packet to packet, and the technique disclosedin Fu works well at high signal-to-noise ratio (SNR).

SUMMARY OF THE INVENTION

To overcome the above-mentioned drawbacks, a Maximum-Likelihood (ML)estimation technique is proposed, which works in the frequency domain.It may be difficult to calculate analytically the time value thataddresses the ML drawbacks, and a numerical approach might be tooonerous to be implemented in a real-time demodulation.

An algorithm is proposed in order to significantly reduce thecomputational load without a sensible accuracy loss. The algorithm maybe subdivided in two steps: first, a coarse estimation of the differenceof the synchronization offset is obtained, then a refined estimation iscalculated by processing values in a neighborhood of the coarseestimation.

According to an embodiment, this refined approximation is obtained witha search algorithm. According to another embodiment, this refinedapproximation is obtained with a quadratic Taylor's series approximationabout the coarse estimation.

The algorithm may be applied in OFDM systems and may be implementedthrough hardware, or through software code executed by a processor. AOFDM receiver implementing the method is also proposed.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is an exemplary graph of the function F({tilde over (τ)}) forτ_(p,p-1)=12.5 at SNR=5 dB of the received signal, in accordance withthe present invention.

FIG. 2 are exemplary graphs of the function d({tilde over (τ)},k) forsub-carriers 1, 2, 3 and 4 with τ_(p,p-1)=12.5 in the absence of noise,in accordance with the present invention.

FIG. 3 is a magnified view of the graph of FIG. 1 F({tilde over (τ)}) ina neighborhood of τ_(p,p-1)=12.5 in accordance with the presentinvention.

FIG. 4 compares the performances in terms of mean square error MSE vs.SNR of the received signal, for the algorithm TS for different values ofW and the maximum likelihood algorithm ML, in accordance with thepresent invention.

FIG. 5 depicts the probability P that the coarse estimate τ′_(p,p-1)lies in a limited neighborhood of τ_(p,p-1) for different values of thewidth W of the neighborhood as a function of the SNR of the receivedsignal, in accordance with the present invention.

FIG. 6 compares the BER performances vs. SNR for a B-PSK modulationusing the ML algorithm, the algorithm TS, the prior algorithms SHE, HE,and XTD with the OPERA 1 channel model and AWGN and a lower bound forB-PSK transmissions, in accordance with the present invention.

FIG. 7 compares the BER performances vs. SNR for a 1024-QAM modulationusing the ML algorithm, the algorithm TS, the prior algorithms SHE, HE,and XTD with the OPERA 1 channel model and AWGN and a lower bound for1024-QAM transmissions, in accordance with the present invention.

FIG. 8 is a block diagram of a receiver that implements a method inaccordance with the present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

In order to better understand the field of the novel method, a briefreview of OFDM is presented. The mathematical analysis of the methodwill be carried out based upon the hypothesis that the phase carrieroffset and the frequency carrier offset are null. This condition isrigorously verified in base-band communications and is practicallyverified in a receiver that has dedicated circuits for compensatingcarrier offsets.

These dedicated circuits are well known in the practice and commonlyused in receivers that need them. For this reason, it may be fairlyassumed in the ensuing description that carrier offsets are negligible.

In communication systems in which this hypothesis is not verified, themethod is still usable, though performances may be worse than thoseillustrated in this description and may depend on the amplitude ofcarrier offsets.

In OFDM-based packet communication systems, the data symbols areserial-to-parallel converted into N streams and fed into the OFDMmodulator implemented by using a N-point inverse discrete Fouriertransform (IDFT). Successively, a Cyclic Prefix (CP) is added at thebeginning of each OFDM symbol to mitigate inter-symbol and inter-carrierinterference (ISI and ICI). At the receiver, the signal is sampled andthe samples sent to a frame synchronization block. After the CP removal,OFDM demodulation is performed by means of a N-point discrete Fouriertransform (DFT). Through this work the following assumptions are made:the channel is time-invariant within several packets, and a smallsampling frequency offset between the transmitter and the receiverclocks is present.

However, even a small sampling frequency offset between the transmitterand the receiver clocks leads to a Sampling Phase Offset (SPO), which isdifferent from packet to packet. Hereinafter Δ_(n) ^(SPO) designates theSPO between the transmitter and the receiver clocks during the n-thpacket. With respect to the frame synchronization block, in the absenceof noise, it is expected to estimate the same synchronization point ineach packet.

Unfortunately, due to the noise, the estimate of the beginning of thepacket may differ from the expected synchronization point and a FrameSynchronization Offset (FSO) occurs. Hereinafter Δ_(n) ^(FSO) designatesthe FSO occurred during the n-th packet.

Let H(k) be the channel frequency response complex coefficient over thek-th sub-carrier. Let X_(n,s)(k) and W_(n,s)(k) be the frequency domaintransmitted symbol and the additive noise, eventually comprising theresidual ISI and ICI, respectively, over the k-th sub-carrier for thes-th OFDM symbol during the n-th packet. The received sample at the DFToutput is

$\begin{matrix}{{{Y_{n,s}(k)} = {{{H(k)}{\mathbb{e}}^{{\mathbb{i}}\frac{2\pi\; k}{N}\Delta_{n}}{X_{n,s}(k)}} + {W_{n,s}(k)}}}{where}} & (1) \\{\Delta_{n} = {\Delta_{n}^{SPO} + \Delta_{n}^{FSO}}} & (2)\end{matrix}$is the synchronization offset, N is the number of points of the discreteFourier transform, Δ_(n) ^(SPO) is a fraction of the sampling time,Δ_(n) ^(FSO) is an integer multiple, positive or negative, of thesampling time, and both may be different from packet to packet. Theestimated transmitted sample, here denoted as {circumflex over(X)}_(n,s)(k), is obtained using a one-tap equalizer as follows

$\begin{matrix}{{{\hat{X}}_{n,s}(k)} = \frac{Y_{n,s}(k)}{{H(k)}{\mathbb{e}}^{{\mathbb{i}}\frac{2\pi\; k}{N}\Delta_{n}}}} & (3)\end{matrix}$

It should be understood that the term

${H(k)}{\mathbb{e}}^{{\mathbb{i}}\frac{2\pi\; k}{N}\Delta_{n}}$is not known at the receiver, but it has to be estimated.

A data-aided channel estimation may be performed during the currentpacket by exploiting the known symbols within the header. Using a LS(Least Square) estimator, as discussed in, “On channel estimation inOFDM systems,” IEEE VTC 1995, pp. 815-819, July 1995, J.-J van de Beek,O. Edfors, M. Sandell, S. K. Wilson and P. O. Borjesson, over S symbolsof the header, the channel estimation performed over the k-thsub-carrier during the n-th packet is

$\begin{matrix}{{{\hat{H}}_{n}(k)} = {\frac{1}{S}{\sum\limits_{s = 1}^{S}\;\frac{Y_{n,s}(k)}{X_{n,s}(k)}}}} & (4)\end{matrix}$

Substituting (1) into (4), the estimate Ĥ_(n)(k) is

$\begin{matrix}{{{\hat{H}}_{n}(k)} = {{{H(k)}{\mathbb{e}}^{{\mathbb{i}}\frac{2\pi\; k}{N}\Delta_{n}}} + {N_{n}(k)}}} & (5) \\{where} & \; \\{{N_{n}(k)} = {\frac{1}{S}{\sum\limits_{s = 1}^{S}\frac{W_{n,s}(k)}{X_{n,s}(k)}}}} & (6)\end{matrix}$is the noise with variance σ_(N) ²(k).

The accuracy of the channel estimation available during the p-th packetmay be improved by averaging out the channel estimates (5) collected upto the p-th packet taking into account that the differences among thesynchronization offsets of different packets may be known. Let τ_(p,n)be the difference between the synchronization offset of the p-th andn-th packets, defined asτ_(p,n)=Δ_(p)−Δ_(n).  (7)

The averaged channel estimation performed over the k-th sub-carrierduring the p-th packet is

$\begin{matrix}{{{\overset{\_}{H}}_{p}(k)} = {\frac{1}{p}{\sum\limits_{n = 1}^{p}{{{\hat{H}}_{n}(k)}{\mathbb{e}}^{{\mathbb{i}}\frac{2\pi\; k}{N}\tau_{p,n}}}}}} & (8)\end{matrix}$

Substituting (5) into (8), the estimate H _(p)(k) is

$\begin{matrix}{{{\overset{\_}{H}}_{p}(k)} = {{{H(k)}{\mathbb{e}}^{\iota\frac{2\pi\; k}{N}\Delta_{p}}} + {{\overset{\_}{N}}_{p}(k)}}} & (9) \\{where} & \; \\{{{\overset{\_}{N}}_{p}(k)} = {\frac{1}{p}{\sum\limits_{n = 1}^{p}{{N_{n}(k)}{\mathbb{e}}^{{\mathbb{i}}\frac{2\pi\; k}{N}\tau_{p,n}}}}}} & (10)\end{matrix}$is the noise with variance σ _(N) _(p) ²(k)=σ_(N) ²(k)/p. In order toreduce the computational effort, equation (8) may be rewritten in thefollowing recursive form

$\begin{matrix}{{{\overset{\_}{H}}_{p}(k)} = \left\{ \begin{matrix}{{{\hat{H}}_{1}(k)},} & {p = 1} \\{\frac{{{\hat{H}}_{p}(k)} + {\left( {p - 1} \right){{\overset{\_}{H}}_{p - 1}(k)}{\mathbb{e}}^{{\mathbb{i}}\frac{2\pi\; k}{N}\tau_{p,{p - 1}}}}}{p},} & {p > 1}\end{matrix} \right.} & (11)\end{matrix}$where the estimate performed during the p-th packet, Ĥ_(p)(k), is usedto update the estimate accumulated during the p−1 previous packets, H_(p-1)(k).

In order to properly add Ĥ_(p)(k) to H _(p-1)(k) in (11), τ_(p,p-1) maybe estimated. To this aim, the ML (Maximum Likelihood) estimator isproposed. From (5) and (9), it is possible to relate τ_(p,p-1) withĤ_(p)(k) and H _(p-1)(k) as follows:

$\begin{matrix}{{{\hat{H}}_{p}(k)} = {{{{\overset{\_}{H}}_{p - 1}(k)}{\mathbb{e}}^{{\mathbb{i}}\frac{2\pi\; k}{N}\tau_{p,{p - 1}}}} + {N_{p}^{\prime}(k)}}} & (12) \\{where} & \; \\{{N_{p}^{\prime}(k)} = {{N_{p}(k)} - {{{\overset{\_}{N}}_{p - 1}(k)}{\mathbb{e}}^{{\mathbb{i}}\frac{2\pi\; k}{N}\tau_{p,{p - 1}}}}}} & (13)\end{matrix}$is the noise with variance σ_(N′) _(p) ²(k)=σ_(N) ²(k)+σ_(N) ²(k)/(p−1).Starting from (12) and assuming N′_(p)(k) to be Gaussian noise with zeromean, the ML estimator, as discussed in S. M. Kay, Fundamentals ofStatistical Processing, Volume I: Estimation Theory. Prentice Hall,1993, of τ_(p,p-1) is

$\begin{matrix}{{\hat{\tau}}_{p,{p - 1}} = {\arg\;{\min\limits_{\overset{\sim}{\tau} \in \Gamma}{\sum\limits_{k \in \Psi}{\frac{1}{\sigma_{N_{p}^{\prime}}^{2}(k)}{d\left( {\overset{\sim}{\tau},k} \right)}}}}}} & (14)\end{matrix}$where Γ is the set of values that τ_(p,p-1) may assume, Ψ is a sub-setof the sub-carriers used in the OFDM system and

$\begin{matrix}{{d\left( {\overset{\sim}{\tau},k} \right)} = {{{{\hat{H}}_{p}(k)} - {{{\overset{\_}{H}}_{p - 1}(k)}{\mathbb{e}}^{{\mathbb{i}}\frac{2\pi\; k}{N}\overset{\sim}{\tau}}}}}^{2}} & (15)\end{matrix}$is the square of the Euclidean distance between Ĥ_(p)(k) and

${{\overset{\_}{H}}_{p - 1}(k)}{{\mathbb{e}}^{{\mathbb{i}}\frac{2\pi\; k}{N}\overset{\sim}{\tau}}.}$The ML estimator may be intuitively explained as follows: consideringthe Gauss plane, if {tilde over (τ)} changes the vector

${{\overset{\_}{H}}_{p - 1}(k)}{\mathbb{e}}^{{\mathbb{i}}\frac{2\pi\; k}{N}\overset{\sim}{\tau}}$rotates and the Euclidean distance d({tilde over (τ)},k) changes.d({tilde over (τ)},k) is minimum when the vectors Ĥ_(p)(k) and

${{\overset{\_}{H}}_{p - 1}(k)}{\mathbb{e}}^{{\mathbb{i}}\frac{2\pi\; k}{N}\overset{\sim}{\tau}}$and overlap. In the absence of noise, the estimate {circumflex over(τ)}_(p,p-1) is the value of {tilde over (τ)} that simultaneouslyminimizes d({tilde over (τ)},k) for each k in Ψ. The sum in (14) is aperiodic function with period N, hence the estimator may notdiscriminate between τ_(p,p-1) and τ_(p,p-1)+lN, with/integer. Supposingthat the frame synchronization algorithm properly works, it isreasonable to assume that τ_(p,p-1) lies in Γ=[−N/2,N/2).

A drawback of the ML estimator is its computational complexity. However,after some manipulations, (14) may be simplified as

$\begin{matrix}{{\hat{\tau}}_{p,{p - 1}} = {\arg\;{\max\limits_{\overset{\sim}{\tau} \in \Gamma}{F\left( \overset{\sim}{\tau} \right)}}}} & (16) \\{where} & \; \\{{F\left( \overset{\sim}{\tau} \right)} = {\sum\limits_{k \in \Psi}{f\left( {\overset{\sim}{\tau},k} \right)}}} & (17) \\{and} & \; \\{{f\left( {\overset{\sim}{\tau},k} \right)} = {{{A_{p}(k)}{\cos\left( {\frac{2\pi\; k}{N}\overset{\sim}{\tau}} \right)}} + {{B_{p}(k)}{\sin\left( {\frac{2\pi\; k}{N}\overset{\sim}{\tau}} \right)}}}} & (18) \\{with} & \; \\{{A_{p}(k)} = {\frac{1}{\sigma_{{\overset{\_}{N}}_{p}}^{2}(k)}\left\{ {{{\Re\left\lbrack {{\hat{H}}_{p}(k)} \right\rbrack}{\Re\left\lbrack {{\overset{\_}{H}}_{p - 1}(k)} \right\rbrack}} + {{{??}\left\lbrack {{\hat{H}}_{p}(k)} \right\rbrack}{{??}\left\lbrack {{\overset{\_}{H}}_{p - 1}(k)} \right\rbrack}}} \right\}}} & (19) \\{and} & \; \\{{B_{p}(k)} = {\frac{1}{\sigma_{{\overset{\_}{N}}_{p}}^{2}(k)}\left\{ {{{{??}\left\lbrack {{\hat{H}}_{p}(k)} \right\rbrack}{\Re\left\lbrack {{\overset{\_}{H}}_{p - 1}(k)} \right\rbrack}} - {{\Re\left\lbrack {{\hat{H}}_{p}(k)} \right\rbrack}{{??}\left\lbrack {{\overset{\_}{H}}_{p - 1}(k)} \right\rbrack}}} \right\}}} & (20)\end{matrix}$wherein

[.] and τ[.] are the real part and the imaginary part of their argument,respectively.

FIG. 1 shows a realization of F({tilde over (τ)}) for τ_(p,p-1)=12.5 andN=384, with an additive white Gaussian noise (AWGN) at a signal-to-noiseratio (SNR), defined hereinafter, equal to 5 dB.

Although the shape of F({tilde over (τ)}) depends on the channel, on thenoise and on the value of τ_(p,p-1), in the range Γ the functionF({tilde over (τ)}) presents one absolute maximum, which corresponds to{circumflex over (τ)}_(p,p-1) and several local maxima. The value of{tilde over (τ)} that maximizes F({tilde over (τ)}) may not be foundanalytically. Moreover, a maximum search algorithm may be useless in thewhole range Γ because of the presence of several local maxima.

This value may be found by iteratively calculating F({tilde over (τ)})for discrete values of {tilde over (τ)}. Let U be the cardinality of Ψand let T be the number of discrete values of {tilde over (τ)}(iterations) for which F({tilde over (τ)}) is calculated. Thecomputational complexity in terms of U and T is reported in Table 1. Thecomputational load related to the evaluation of the 2πk/N terms is notconsidered in the computational complexity, since these terms areconsidered constants for a given OFDM system. The accuracy of theestimate depends on the resolution of the discrete values of {tilde over(τ)}, which increases with T.

TABLE 1 Computational complexity of the ML estimator by iterativesearch. Cosine & Sine 2UT Additions 2U + (2U − 1)T Multiplications 6U +3UT Divisions U

The above calculations are relatively onerous to be executed in areceiver. The complexity of the estimator (16) is reduced by:

i) finding a coarse estimate of τ_(p,p-1), hereafter referred asτ_(p,p-1)′;

ii) improving the coarse estimate to find an accurate estimate ofτ_(p,p-1), hereafter referred as τ _(p,p-1).

Regarding the first step, let α_(p,p-1)(k)ε[−π,π) be the phasedifference between Ĥ_(p)(k) and H _(p-1)(k). The distance functiond({tilde over (τ)},k), defined in Equation (15), is minimized whenĤ_(p)(k) and

${{\overset{\_}{H}}_{p - 1}(k)}{\mathbb{e}}^{{\mathbb{i}}\frac{2\pi\; k}{N}\overset{\sim}{\tau}}$overlap, that is when

$\begin{matrix}{{\frac{2\pi\; k}{N}\overset{\sim}{\tau}} = {{\alpha_{p,{p - 1}}(k)} + {2\pi\; l}}} & (21)\end{matrix}$

being l an integer. Considering the k-th sub-carrier, within the range Γthe condition (21) is satisfied for k values of {tilde over (τ)},referred as {tilde over (τ)}_(l)(k), and given by

$\begin{matrix}{{{\overset{\sim}{\tau}}_{l}(k)} = {{\frac{N}{2\pi\; k}{\alpha_{p,{p - 1}}(k)}} + {\frac{N}{k}l}}} & (22)\end{matrix}$with l=0, 1, . . . , k−1. When {tilde over (τ)}_(l)(k) is greater thanN/2, it is remapped in the range Γ by subtracting N. In general, eachfunction d({tilde over (τ)},k) has a minimum at {tilde over (τ)}_(l)(k)with l=0, 1, . . . , k−1. However, in the absence of noise, there may beonly one {tilde over (τ)}_(l)(k)=τ_(p,p-1) at which all functionsd({tilde over (τ)},k) have a minimum for any sub-carrier k. To clarifythis assertion, in FIG. 2 the graphs of the function d({tilde over(τ)},k) are shown for k=1,2,3 and 4, in the absence of noise.

Based on this consideration, a coarse estimate of τ_(p,p-1) may beobtained by crosschecking all {tilde over (τ)}_(l)(k) for allsub-carriers by means of a histogram.

In particular, the interval Γ has been divided into N/W sub-intervals ofwidth W, called histogram bin size, centered in {tilde over (τ)}_(i)=Wi,with i=−N/2W, . . . , N/2W. The coarse estimate τ_(p,p-1)′ is given bythe center of the sub-interval where the histogram has its maximumvalue. If noise is present, no {tilde over (τ)}_(l)(k) exactly matchesτ_(p,p-1), but the density of {tilde over (τ)}_(l)/(k) is statisticallyhigher in the sub-interval where τ_(p,p-1) lies, rather than in othersub-intervals.

As regards the second step to improve the coarse estimate, we highlightthat a zoom of F({tilde over (τ)}) in a neighborhood of τ_(p,p-1) showsthat the curve is concave and that only one maximum is present, asdepicted in FIG. 3.

A solution may be to apply a maximum search algorithm in a limitedneighborhood of τ_(p,p-1)′, where F({tilde over (τ)}) is still concaveif the distance between and τ_(p,p-1)′ and τ_(p,p-1) is small. In thiscase, the maximum search algorithm may converge to τ_(p,p-1) in fewiterations.

An alternative solution, which further reduces the computational load,is approximating F({tilde over (τ)}) by means of the Taylor series ofdegree 2 about τ_(p,p-1)′ and setting the first derivative of the seriesequal to zero. The refined estimation is given by

$\begin{matrix}{{\overset{\_}{\tau}}_{p,{p - 1}} = {\tau_{p,{p - 1}}^{\prime} - \frac{{\sum\limits_{k \in \Psi}{{A_{p}(k)}{\sin\left( {\frac{2\pi\; k}{N}\tau_{p,{p - 1}}^{\prime}} \right)}}} - {{B_{p}(k)}{\cos\left( {\frac{2\pi\; k}{N}\tau_{p,{p - 1}}^{\prime}} \right)}}}{\sum\limits_{k \in \Psi}{\frac{2\pi\; k}{N}\left\lbrack {{{A_{p}(k)}{\cos\left( {\frac{2\pi\; k}{N}\pi_{p,{p - 1}}^{\prime}} \right)}} + {{B_{p}(k)}{\sin\left( {\frac{2\pi\; k}{N}\tau_{p,{p - 1}}^{\prime}} \right)}}} \right\rbrack}}}} & (23)\end{matrix}$

The accuracy of (23) depends on the closeness of τ_(p,p-1)′ toτ_(p,p-1), hence it increases when the histogram bin size W decreases.However, if W is too small, the histogram used to estimate τ_(p,p-1)′,due to the noise, does not properly work and the probability ofτ_(p,p-1)′ being far from τ_(p,p-1) increases. Therefore, a goodtrade-off to properly choose W has to be found.

According to an embodiment, W is pre-established. According to anotherembodiment, W is determined heuristically depending on the channelcharacteristics.

The bin size W may also be determined using a procedure disclosed indetail hereinafter.

FIG. 8 depicts a block diagram of a receiver that implements the method.The computational complexity for the search of τ_(p,p-1)′ and theapplication of (23) are shown in Tables 2 and 3, respectively. Table 2does not include the computational load of the histogram. This methodwill be referred to as Taylor series approximation (TS).

TABLE 2 Computational complexity for the search of τ′_(p,p−1). Phase Udifferences Additions Σ_(keΨ)(k − 1) Multiplications U

TABLE 3 Computational complexity of the estimator based on the Taylorseries. Cosine & Sine 2U Additions 2U − 1 Multiplications 12U DivisionsU + 1To analyze the performance of the proposed algorithm, the HomePlug AV(HPAV) system HomePlug PowerLine Alliance, “HomePlug AV specification,”May 2007, version 1.1., “HomePlug AV white paper,”http://www.homeplug.org has been chosen, and a power-line environmenthas been simulated considering the channel models proposed by the openpower-line communication European research alliance (OPERA) project“Theoretical postulation of PLC channel model,” M. Babic, M. Hagenau, K.Dostert and J. Bausch March 2005. tech. Rep., OPERA. Through numericalresults, the estimator based on the Taylor series is compared with theML estimator, showing nearly the same performance. Moreover, both areshown to achieve the performance of an ideal system where the phasedifferences among the packets are known at the receiver.HPAV System and Channel Model

HPAV is a suitable system to analyze the performances of the proposedalgorithm since neither phase nor frequency carrier offset is present.Furthermore, HPAV provides a sounding procedure that includes sendingconsecutive packets known at the receiver to probe the characteristicsof the channel before establishing a new connection. To improve thechannel estimation, the sounding procedure may be exploited by averagingout the channel estimates performed during the packets of the sounding.This is possible when the differences among the synchronization offsetsof different packets are estimated. In a typical HPAV packet, the knownsymbols of the header are followed by the information data of thepayload. During the sounding procedure also the payload is known at thereceiver. The channel estimation in (11) is applied both among soundingpackets and among sounding and data packets. In particular, during thesounding, (4) is applied to the payload (S=20), while in the data packet(4) may be applied to the header only (S=4). Applicants have observedthat, in tests, the sounding procedure exploits 4 packets. The mainparameters used in the simulations are resumed in Table 4.

TABLE 4 Parameters used in the simulations. HPAV Modulation order B-PSK,1024-QAM Turbo code rate ½ FFT-points (N) 3072 Symbols of the header 4Symbols of the payload during 20 sounding Packets of sounding 4

As regards the channel, a power-line environment has been modeled usingthe channel references for in-house networks proposed by OPERA. Thepresented results are obtained with the multi-path channel model 1,characterized by 5 paths and the impulse response duration of 0.5 μs.For further details, the interested reader should refer to M. Babic etal. AWGN noise is included.

TS Algorithm Performance

Performance of novel TS algorithm is analyzed in the presence of SPO andFSO changing from packet to packet and for different SNR values, wherethe SNR is defined as the ratio between the power of the received signaland the power of the noise over the signal bandwidth. To implement theTS algorithm, the following parameters may be defined:

1) the number of sub-carriers used, U;

2) the histogram bin size, W.

If the number of sub-carriers is great, the statistical of the histogramis more accurate and the coarse estimate results more robust to thenoise. On the other hand, the complexity of the algorithm increases, asdescribed in Table 2 and Table 3.

In the following simulations, U is set equal to 100, since this valueresults in a good trade-off between performance and algorithmcomplexity. As far as the bin size W of the histogram is concerned, anexemplary way of choosing this value according to the channelcharacteristics is presented herein below. One or more symbols of theheader is/are transmitted and the following analysis in an AWGN channelis carried out:

1. Let L be a positive real number and let τ_(p,p-1)′ be chosen suchthat |τ_(p,p-1)′−τ_(p,p-1)= L. The refined estimations τ _(p,p-1) arecalculated for different realizations of noise by means of Equation(23).

2. The mean square error (MSE), defined asMSE=E{| τ _(p,p-1)−τ_(p,p-1)|²}  (24)is calculated, where E(x) denotes the expectation of x.

3. Step 1. and 2. are repeated, at the SNRs of interest, for differentvalues of L.

4. The value L is chosen as the biggest value of L that allows reachinga desired MSE.

5. Let the histogram bin size be equal to W. The value τ_(p,p-1)′ isdetermined as the value that maximizes the histogram. The probability P,defined asP=Pr{|τ _(p,p-1)′−τ_(p,p-1) |<L}  (25)is calculated.

6. Step 5. is repeated for different values of W.

7. The value W is chosen as the value of W that maximizes P.

The MSE and the probability P are shown in FIG. 4 and FIG. 5,respectively, over an AWGN channel when only one symbol of the header isused for the channel estimation (S=1). In particular, in FIG. 4 the MSEfor different values of L from 1/16 to ½ is shown. For comparison, theMSE of the ML estimator is also reported. Performance improves as Ldecreases, almost overlapping the ML estimator for L≦⅙, hence L=⅙ ischosen and it is used in the calculus of the probability P. FIG. 5 showsthe probability P for different values of W; initially, P improves as Wdecreases, but the trend of P inverts for the smallest values of W. Fromthese considerations, the value W=¼ is adopted for the remainingsimulations.

Performances of the TS algorithm, applied to the described system, areanalyzed in terms of bit error rate (BER) versus SNR. In order tocompare the proposed algorithms to other solutions, the following curvesare also reported:

bound: the channel estimation is performed exploiting the packets of thesounding, and the synchronization offset is assumed known at thereceiver. This represents the bound that the algorithm may achieve.

sound+header estimation (SHE): the estimate of the channel magnitude isperformed exploiting all the packets of the sounding, but the estimateof the phase is performed using the header of the current data packet.Indeed, the magnitudes are not affected by synchronization offsets, thusthey may be correctly averaged. On the other hand, estimating the phaseover the data packet allows proper absorption of the currentsynchronization offset.

header estimation (HE): the channel estimation is performed using theheader of the current data packet. This is the coarser channelestimation, which does not exploit the sounding at all.

cross-correlation in time domain (XTD): the channel estimation isperformed by means of the algorithm based on the time-domaincross-correlation proposed in Riva at al.

FIG. 6 and FIG. 7 depict the BER obtained for different SNRs when theB-PSK and 1024-QAM, respectively, are adopted. In particular, FIG. 6shows that the ML and TS solutions almost overlap the bound.

Considering a BER of 10⁻³, a gain of about 0.6 dB and of more than 1.5dB is present compared to the SHE and the HE estimators, respectively.The poor performance of the XTD proves the sensibility of this algorithmto the SPO changes. On the other hand, when the SPO is constant frompacket to packet, the XTD is comparable to the system performance bound,as shown in [1].

Similar results are illustrated in FIG. 7, where the bound, the ML andthe TS are almost coincident, and they gain almost 0.4 dB and 0.9 dBcompared to the SHE and the HE estimators, respectively, at a BER=10⁻³.at a. The detrimental effects of the SPO on the XTD are even moreevident for the 1024-QAM modulation, which is less robust to phaserotations. It has to be noticed that the ML and the TS estimatorsachieve the same performance, but their computational load is verydifferent. In particular, the B-PSK modulation requires that thedifferences among the synchronization offsets be estimated with aresolution of at least 10⁻². Furthermore, the employed framesynchronization algorithm guarantees that the differences among thesynchronization offsets are within the interval [−20,20] for all theSNRs of interest. Therefore, the number of iterations, T, of the ML isequal to 40/10⁻²=4·10³, while U is 100. Referring to Table 1, about8·10⁵ additions and more than 1.2·10⁶ multiplications are required forthe ML. In the same conditions, the TS requires only about 9·10³additions and 1.3·10³ multiplications.

The invention claimed is:
 1. A method of estimating transmission channelresponse and differences of synchronization offsets in a received streamof packets of OFDM data transmitted on a transmission channel, thestream of OFDM data being preceded by a header of symbols, the methodcomprising: performing, using a processor, a data-aided estimation of achannel response using the header when a packet is received, to therebygenerate a corresponding estimated channel response value Ĥ_(p)(k) foreach sub-carrier (k) used in OFDM modulation; for a first receivedpacket, storing the estimated channel response Ĥ_(l)(k) as a refinedchannel response H _(l)(k) for each of the sub-carriers (k), using theprocessor; for successive packets, using the processor to iterativelycalculate a phase difference α_(p,p-1)(k) between the estimated channelresponse Ĥ_(p)(k) for a current packet (p) and a refined channelresponse for a previous packet H _(p-1)(k) for each of the sub-carriers(k), estimate values of differences of synchronization offsets {tildeover (τ)}_(l)(k) for each phase difference α_(p,p-1)(k) for each of thesub-carriers (k), choose a set of quantized values for each differenceof synchronization offsets, choose, in the set of quantized values, acoarse estimation of the difference of synchronization offsets(τ′_(p,p-1)) for the current packet as a quantized value that maximizesa histogram of synchronization offsets {tilde over (τ)}_(l)(k) versusquantized values, process values of a neighborhood of the coarseestimation (τ′_(p,p-1)) with an algorithm configured to calculate arefined difference of synchronization offsets for the current packet asa value that minimizes a sum of the Euclidean distances, for frequenciesof the sub-carriers, between the estimated channel response Ĥ_(p)(k) andthe refined channel response H _(p-1)(k) rotated in a complex plane by aphase offset proportional to the refined difference of synchronizationoffsets, and calculate the refined channel response for the currentpacket H _(p)(k) for each of the carriers (k) as a weighted average of anumber p of received packets, of a rotated replica of the refinedchannel response H _(p-1)(k) multiplied by p−1 and the estimated channelresponse Ĥ_(p)(k).
 2. The method of claim 1, wherein the algorithmcomprises an iterative search algorithm using, as a seed, the coarseestimation (τ′_(p,p-1)).
 3. The method of claim 1, wherein the algorithmis based on a quadratic Taylor's series approximation of the sum ofEuclidean distances in the neighborhood of the coarse estimation(τ′_(p,p-1)).
 4. The method of claim 1, wherein a bin size of thehistogram is pre-established.
 5. The method of claim 1, wherein a binsize of the histogram is determined by at least: determining thehistogram bin size as a value that maximizes a probability that adistance between a difference of synchronization offset from the coarseestimation be smaller than a maximum error of the coarse estimation. 6.The method of claim 5, wherein the maximum error of the coarseestimation is determined by at least: determining a mean square error ofthe refined estimation for which the system achieves a performance;determining the maximum error of the coarse estimation as the value thatresults in mean square error.
 7. A method of estimating transmissionchannel response and differences of synchronization offsets in areceived stream of packets of OFDM data transmitted on a transmissionchannel, the stream of OFDM data being preceded by a header of symbols,the method comprising: performing, using a processor, a data-aidedestimation of a channel response using the header when a packet isreceived, to thereby generate a corresponding estimated channel responsevalue for each sub-carrier used in OFDM modulation; for a first receivedpacket, storing the estimated channel response as a refined channelresponse for each of the sub-carriers, using the processor; forsuccessive packets, using the processor to iteratively calculate a phasedifference between the estimated channel response for a current packetand a refined channel response for a previous packet for each of thesub-carriers, estimate values of differences of synchronization offsetsfor each phase difference for each of the sub-carriers, choose a set ofquantized values for each difference of synchronization offsets, choose,in the set of quantized values, a coarse estimation of the difference ofsynchronization offsets for the current packet as a quantized value fora histogram of synchronization offsets versus quantized values, processvalues of a neighborhood of the coarse estimation with an algorithmconfigured to calculate a refined difference of synchronization offsetsfor the current packet as a value for a sum of the Euclidean distances,for frequencies of the sub-carriers, between the estimated channelresponse and the refined channel response rotated in a complex plane bya phase offset based upon the refined difference of synchronizationoffsets, and calculate the refined channel response for the currentpacket for each of the carriers as a weighted average of a number p ofreceived packets, of a rotated replica of the refined channel responsemultiplied by p−1 and the estimated channel response.
 8. The method ofclaim 7, wherein the algorithm comprises an iterative search algorithmusing, as a seed, the coarse estimation.
 9. The method of claim 7,wherein the algorithm is based on a quadratic Taylor's seriesapproximation of the sum of Euclidean distances in the neighborhood ofthe coarse estimation.
 10. An OFDM receiver comprising: a samplingcircuit configured to sample an incoming signal received through atransmission channel; an estimation circuit configured to receivesamples of the incoming signal and to estimate transmission channelresponse and eventual differences of synchronization offsets introducedat a receiver side by at least: performing a data-aided estimation of achannel response using the header when a packet is received, to therebygenerate a corresponding estimated channel response value for eachsub-carrier used in OFDM modulation; for a first received packet,storing the estimated channel response as a refined channel response foreach of the sub-carriers, for successive packets, iterativelycalculating a phase difference between the estimated channel responsefor a current packet and a refined channel response for a previouspacket for each of the sub-carriers, estimating values of differences ofsynchronization offsets for each phase difference for each of thesub-carriers, choosing a set of quantized values for each difference ofsynchronization offsets, choosing, in the set of quantized values, acoarse estimation of the difference of synchronization offsets for thecurrent packet as a quantized value for a histogram of synchronizationoffsets versus quantized values, processing values of a neighborhood ofthe coarse estimation with an algorithm configured to calculate arefined difference of synchronization offsets for the current packet asa value for a sum of the Euclidean distances, for frequencies of thesub-carriers, between the estimated channel response and the refinedchannel response rotated in a plane by a phase offset based upon therefined difference of synchronization offsets, and calculating therefined channel response for the current packet for each of the carriersas a weighted average of a number p of received packets, of a rotatedreplica of the refined channel response multiplied by p−1 and theestimated channel response.
 11. The OFDM receiver of claim 10, furthercomprising an equalizer coupled to the estimation circuit and configuredto compensate an effect of the transmission channel response and of thedifferences of synchronization offsets on the received samples and togenerate equalized samples.
 12. The OFDM receiver of claim 11, furthercomprising an OFDM detector configured to generate a stream ofdemodulated digital symbols based upon the equalized samples.
 13. TheOFDM receiver of claim 10, wherein the algorithm comprises an iterativesearch algorithm using, as a seed, the coarse estimation.
 14. The OFDMreceiver of claim 10, wherein the algorithm is based on a quadraticTaylor's series approximation of the sum of Euclidean distances in theneighborhood of the coarse estimation.
 15. The OFDM receiver of claim10, wherein a bin size of the histogram is pre-established.
 16. The OFDMreceiver of claim 10, wherein a bin size of the histogram is determinedby at least: determining the histogram bin size as a value thatmaximizes a probability that a distance between a difference ofsynchronization offset from the coarse estimation be smaller than amaximum error of the coarse estimation.
 17. The OFDM receiver of claim16, wherein the maximum error of the coarse estimation is determined byat least: determining a mean square error of the refined estimation forwhich the system achieves a performance; determining the maximum errorof the coarse estimation as the value that results in mean square error.18. A non-transitory computer readable medium storing executableinstructions that, when executed, cause a processor to estimatetransmission channel response and differences of synchronization offsetsintroduced in a received stream of packets of OFDM data transmitted on atransmission channel, the stream of OFDM data being preceded by a headerof symbols, by at least: performing a data-aided estimation of a channelresponse using the header when a packet is received, to thereby generatea corresponding estimated channel response value for each sub-carrierused in OFDM modulation; for a first received packet, storing theestimated channel response as a refined channel response for each of thesub-carriers; for successive packets, iteratively calculating a phasedifference between the estimated channel response for a current packetand a refined channel response for a previous packet for each of thesub-carriers, estimating values of differences of synchronizationoffsets for each phase difference for each of the sub-carriers, choosinga set of quantized values for each difference of synchronizationoffsets, choosing, in the set of quantized values, a coarse estimationof the difference of synchronization offsets for the current packet as aquantized value for a histogram of synchronization offsets versusquantized values, processing values of a neighborhood of the coarseestimation with an algorithm configured to calculate a refineddifference of synchronization offsets for the current packet as a valuefor a sum of the Euclidean distances, for frequencies of thesub-carriers, between the estimated channel response and the refinedchannel response rotated in a complex plane by a phase offset based uponthe refined difference of synchronization offsets, and calculating therefined channel response for the current packet for each of the carriersas a weighted average of a number p of received packets, of a rotatedreplica of the refined channel response multiplied by p−1 and theestimated channel response.
 19. The non-transitory computer readablemedium of claim 18, wherein the algorithm comprises an iterative searchalgorithm using, as a seed, the coarse estimation.
 20. Thenon-transitory computer readable medium of claim 18, wherein thealgorithm is based on a quadratic Taylor's series approximation of thesum of Euclidean distances in the neighborhood of the coarse estimation.21. The non-transitory computer readable medium of claim 18, wherein abin size of the histogram is pre-established.
 22. The non-transitorycomputer readable medium of claim 18, wherein a bin size of thehistogram is determined by at least: determining the histogram bin sizeas a value that maximizes a probability that a distance between adifference of synchronization offset from the coarse estimation besmaller than a maximum error of the coarse estimation.
 23. Thenon-transitory computer readable medium of claim 22, wherein the maximumerror of the coarse estimation is determined by at least: determining amean square error of the refined estimation for which the systemachieves a performance; determining the maximum error of the coarseestimation as the value that results in mean square error.